Thermal Properties of the Inhomogeneous Electron Gas

A variational property of the ground-state energy of an electron gas in an external potential v (r), derived by Hohenberg and Kohn, is extended to nonzero temperatures. It is first shown that in the grand canonical ensemble at a given temperature and chemical potential, no two v (r) lead to the same equilibrium density. This fact enables one to define a functional of the density Fn (r) independent of v (r), such that the quantity = (r) n (r) dr+Fn (r) is at a minimum and equal to the grand potential when n (r) is the equilibrium density in the grand ensemble in the presence of v (r).